Nonlinear theory of scattering by localized potentials in metals

In early work, March and Murray gave a perturbation theory of the Dirac density matrix gamma (r, r) generated by a localized potential V (r) embedded in an initially uniform Fermi gas to all orders in V (r). For potentials sufficiently slowly varying in space, they summed the resulting series for r' = r to regain the Thomas-Fermi density rho(r) proportional to [mu - V(r)](3/2), with mu the chemical potential of the Fermi gas. For an admittedly simplistic repulsive central potential V (r) = \A\ exp(-cr), it is first shown here that what amounts to the sum of the March-Murray series for the s-wave (only) contribution to the density, namely p, (r, A), can be obtained in closed form. Furthermore, for specific numerical values of A and c in this exponential potential, the long-range behaviour of p, (r, it) is related to the zero-potential form of March and Murray, which merely suffers a it-dependent phase shift. This result is interpreted in relation to the recent high density screening theorem of Zaremba, Nagy and Echenique. A brief discussion of excess electrical resistivity caused by nonlinear scattering in a Fermi gas is added; this now involves an off-diagonal local density of states. Finally, for periodic lattices, contact is made with the quantum-mechanical defect centre models of Koster and Slater (1954 Phys. Rev. 96 1208) and of Beeby (1967 Proc. R. Soc. A 302 113), and also with the semiclassical approximation of Friedel (1954 Adv. Phys. 3 446). In appendices, solvable low-dimensional models are briefly summarized.